The Progressivity of Equalization Payments in Federations

We investigate the conditions under which an inequality averse and additively separable welfarist constitution maker would always choose to set up a progressive equalization payments scheme in a federation with local public goods. A progressive equalization payments scheme is defined as a list of per capita net (possibly negative) subsidies - one such net subsidy for every jurisdiction - that are decreasing with respect to jurisdictions per capita wealth. We examine these questions in a setting in which the case for progressivity is a priori the strongest, namely, all citizens have the same utility function for the private and the public goods, inhabitants of a given jurisdiction are all identical, and they are not able to move across jurisdictions. We show that the constitution maker favors a progressive equalization payments scheme for all distributions of wealth and all population sizes if and only if its objective function is additively separable between each jurisdiction’s per capita wealth and number of inhabitants. When interpreted as a mean of order r social welfare function, this condition is shown to be equivalent to additive separability of the individual’s indirect utility function with respect to wealth and the price of the public good. Some implications of this restriction to the case where the individual’s direct utility function is additively separable are also derived

Canonical Representation of Set Functions

The representation of a cooperative transferable utility game as a linear combination of unanimity games may be viewed as an isomorphism between not-necessarily additive set functions on the players space and additive ones on the coalitions space. We extend the unanimity-basis representation to general (infinite) spaces of players, study spaces of games of games which satisfy certain properties and provide some conditions for sigma-additivity of the resulting additive set function (on the space of coalitions). These results also allow us to extend some representations of the Choquet integral from finite to infinite spaces.

On the large deviations of a class of modulated additive processes

We prove that the large deviation principle holds for a class of processes inspired by semi-Markov additive processes. For the processes we consider, the sojourn times in the phase process need not be independent and identically distributed. Moreover the state selection process need not be independent of the sojourn times. We assume that the phase process takes values in a finite set and that the order in which elements in the set, called states, are visited is selected stochastically. The sojourn times determine how long the phase process spends in a state once it has been selected. The main tool is a representation formula for the sample paths of the empirical laws of the phase process. Then, based on assumed joint large deviation behavior of the state selection and sojourn processes, we prove that the empirical laws of the phase process satisfy a sample path large deviation principle. From this large deviation principle, the large deviations behavior of a class of modulated additive processes is deduced. As an illustration of the utility of the general results, we provide an alternate proof of results for modulated L´evy processes. As a practical application of the results, we calculate the large deviation rate function for a processes that arises as the International Telecommunications Union’s standardized stochastic model of two-way conversational speech

Combinatorial Assortment Optimization

Assortment optimization refers to the problem of designing a slate of products to offer potential customers, such as stocking the shelves in a convenience store. The price of each product is fixed in advance, and a probabilistic choice function describes which product a customer will choose from any given subset. We introduce the combinatorial assortment problem, where each customer may select a bundle of products. We consider a model of consumer choice where the relative value of different bundles is described by a valuation function, while individual customers may differ in their absolute willingness to pay, and study the complexity of the resulting optimization problem. We show that any sub-polynomial approximation to the problem requires exponentially many demand queries when the valuation function is XOS, and that no FPTAS exists even for succinctly-representable submodular valuations. On the positive side, we show how to obtain constant approximations under a "well-priced" condition, where each product's price is sufficiently high. We also provide an exact algorithm for $k$-additive valuations, and show how to extend our results to a learning setting where the seller must infer the customers' preferences from their purchasing behavior